Yes, you have to turn in your device IF you want to get a new device on EDGE. You don't have to pay for the monthly service after you completely pay off the device which typically is 24 months. If you want to continue your service with that device, you don't have to turn it in.

Verizon Edge FAQs | Verizon Wireless
Does the Verizon Edge program require an agreement?
Yes, when you participate in the Verizon Edge program, you will enter into a 24-month Verizon Edge Agreement. However, this agreement is different from traditional 24-month contracts because you can upgrade after 30 days. Customers must have a Verizon Wireless service agreement when they purchase under Verizon Edge.Can I pay more than my scheduled monthly Verizon Edge installment?
No, unless you are either seeking to Edge Up to a new device, which requires that at the time of the Edge Up transaction you pay at least 50% of the device following the initial 30 days of the Verizon Edge Agreement; or you are seeking to pay off 100% of your Verizon Edge Agreement, which you may do at any time.What is the Edge Up option under Verizon Edge?
Edge Up allows you to turn in your existing Edge phone and upgrade to a new phone without an upgrade fee.What are the eligibility requirements to Edge Up with Verizon Edge?
To be eligible to Edge Up, you must:

Have had your Edge phone for at least 30 days
Have paid at least 50% of the full retail price of the phone
Return the Edge phone in good working condition with no significant damage, as determined by Verizon WirelessI paid off 100% of my Edge phone. Can I trade it in and use the trade-in value toward the cost of a new phone under a new Verizon Edge Agreement?
Yes. You’ll receive the value for the trade, and you can use the balance to make the first payment of your new phone. However, you may not use the trade-in value to reduce the retail price of the new Edge phone.

I wouldn't touch the Edge Plan with a 20' pole. As with AT&T's Next, they seem to be rip offs. They bank on you giving in and upgrading each year, thereby keeping your phone and reselling it as a refurb. The Edge plan will cost you more. I have 4 phones on Verizon's More Everything Plan. I'm happy with that. Unlimited text and talk, 4 shared gb of data, free international texting, and free cloud storage for $247 per month. Verizon has a new single line plan for $60 a month. Unlimited text and talk with 2 gb data. That's a good deal if your on wifi a lot.